Density estimation and manifold learning are useful in many application domains including computer vision, medical image analysis, text document clustering, the analysis of general multi-dimensional data and others.
Density estimation involves calculating the volume of a probability distribution of a continuous variable where that volume is related to the probability that the variable will take a value in a specified range. For example, the variable may represent the location of a body joint of a person in 3D space. In another example, the variable may represent the probability that an image element depicts part of a body organ of one of a specified number of types. Many other examples are possible where it is required to learn a probability density function which provides a good estimate of some empirically observed data.
Manifold learning involves calculating a mapping for transforming data in a high dimensional space to a lower dimensional space whilst preserving similarity relationships between the data points. A similarity relationship may be for example, a geodesic relationship between data points which are image elements which is a distance which takes into account intensity or other gradients of the image. Other similarity relationships comprise distances or affinities such as Gaussian affinities, Euclidean distances, or other distances. Typically once the data is mapped into the lower dimensional space, computations may be performed in the lower dimensional space directly, in a more efficient manner than would otherwise have been possible.
The embodiments described below are not limited to implementations which solve any or all of the disadvantages of known systems for density estimation and/or manifold learning.